Decay of disturbances in turbulent pipe flow

Abstract

The basic equations for turbulent flows in circular pipes at high Reynolds numbers are derived. At a certain cross section a three-dimensional velocity field is prescribed that differs only slightly from the fully developed state. The axial decay of the disturbances is investigated by solving the basic equations using an asymptotic expansion of the solutions. The resulting system of linear ordinary differential equations leads to eigenvalue problems. Eigenvalues and eigenfunctions are presented. The general solution can be expanded into series of eigensolutions. The decaying flow consists of various flow structures that fall into two categories: the U-type (ring vortex, sourcesink pair, source-sink quadrupole, etc.) and the W-type (single, double and multiple streamwise vortices). Each of these flow structures has its own decay behavior. Swirl (single streamwise vortex) has the lowest decay followed by the source-sink pair. The strength of the individual flow structure is described by means of characteristic parameters that are the coefficients in the series expansions of the solution. Comparisons of the theory with experimental results from the literature show very good agreement. The different dependence of the decay rate on the Reynolds number for flows with and without swirl can be clarified. The theory is valid for smooth as well as for rough pipe walls. © Springer-Verlag Berlin Heidelberg 2005.